Variational compilation of matter-wave dynamics in reconfigurable potentials

Quantum-assisted quantum compilation (QAQC) has emerged as a key capability within variational quantum algorithms (VQAs), enabling a target unitary operation to be learned using a parameterized quantum circuit. For discrete as well as continuous-variable quantum systems, this approach allows the target unitary to be encoded in circuits optimized for depth or other resource constraints. In this work, we present an experimental and theoretical study of QAQC for bosonic systems, implemented with an interacting Bose–Einstein condensate (BEC) in a reconfigurable trapping potential. Within this BEC-based QAQC setting, the learning task is formulated as the minimization of a cost function that quantifies the distinguishability of BEC states evolved under either a target unitary or a parameterized ansatz.

We develop a theoretical framework for QAQC in bosonic systems consisting of N interacting bosons distributed among M orthogonal single-particle modes. Focusing on the two-mode approximation relevant to our experimental demonstration, we show that the cost function evaluated from the full quantum dynamics is equivalent to that obtained from Gross–Pitaevskii equation (GPE) simulations. Using numerical simulations, we further investigate how the evolution time and the choice of training set affect the convergence of the variational optimization. We also establish covering-number-based generalization bounds that quantify how the performance of the learned dynamics depends on the number of atoms and the number of training states. Experimentally, we demonstrate the preparation of relevant training states for the two-mode system using an interacting BEC in a reconfigurable potential forming tunable double-well configurations. The system is operated in a strongly interacting regime, as evidenced by the observation of self-trapping under appropriate conditions, thereby validating key assumptions underlying both the theoretical framework and the GPE-based description. Together, these results demonstrate the experimental feasibility of essential components of BEC-based QAQC and lay the groundwork for future fully variational implementations.